You could try to replace the screw, I am assuming Its a 1998 model since the 2012 model has two screws. I advice to replace the screw, or you could try to replace or interchange the battery pack door. If its a 2005 model, I would recommend getting a new battery cover.
That is a 1998 furby battery pack, the two metal contacts on the side touches the other terminals to make a signal, so make sure their intact, and clean any corrosion.
This is a 2005 furby battery door, the screw easily gets stripped, and broken. The corrosion is harder to clean since the battery pack is so narrow. They sell replacements on etsy: https://www.etsy.com/listing/1091061818/...
Hope this helps!
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@cadkins post some pictures of all of this with your QUESTION. Let's see if we can figure out while it loses contact when you mount the cover. Adding images to an existing question
Of course, you could also contact the Toy Expert on here https://www.ifixit.com/User/About/350051... and see if she has another great solution.
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